What is a Binary Number?

This question could also have been asked as What is the Binary Notation for a number? or What is the Binary Number System?

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The Binary Notation is one of the five numerical notations discussed in Chapter 2 of Sir Roger Penrose's book The Emperor's New Mind. The other four notations are Unary, Denary, Expanded Binary, and Machine State. The last one is not so much a numerical notation as it is the Transition Function for a Turing Machine, but it can be converted to any one of the other notations, and for that reason it can also be thought of as a numerical notation.

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Any non-negative integer can be expressed using the Binary Notation, and using the same idea as is used in the Denary (or Decimal) Notation. The difference between these two is that while the Denary uses base 10, the Binary uses base 2. This also means that any number can be expressed in the Binary Notation using only two symbols: 0 and 1.

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There are many excellent videos on YouTube that explain Binary Numbers very nicely. All you need to do is go to www.YouTube.com and then type binary in the search box, and you'll have a generous selection of videos to watch. Here is one from the Khan Academy.

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Here are some examples of how numbers can be expressed in the Binary Notation:

0 is written as 0
1 is written as 1
2 is written as 10
3 is written as 11
4 is written as 100
5 is written as 101

And so on.

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Also, there may be leading 0's with the number, and its value does not change.

For Example:

110, 0110, 00110, and 000110, are all representations of the same number 6.

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Of the five numerical notations, only the Unary and Expanded Binary can be used to represent data on the paper tape that's used by the type of Turing Machine described in Roger Penrose's book The Emperor's New Mind. (The only exception is the Binary Notation, and that's only when it represents the Turing Machine Number for use by the Universal Turing Machine.)

With regard to Penrose's Turing Machine, the Binary and Denary are primarily used when a number is converted from one of the other three notations for a better understanding or recognition by humans.

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